C₂×C₅×#2215 #11956317

order	1280
nilpotency class	4
isoclinism	isoclinic class of #2215
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁶

Aliases

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

806 conjugacy classes of subgroups.

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 640

A minimal faithful permutation representation — 4 generators on 27 points:

(1,2)(23,24,25,26,27);(3,4)(5,6)(8,12)(9,13)(10,16)(14,22)(15,20)(17,21)(18,19);(7,8,13,19,10,14,17,20,11,15,21,22,16,18,9,12);(4,6)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1280, 26585);